Article
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Cogging torque reduction based on a new pre-slot
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technique for a small wind generator
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Miguel García-Gracia 1*, Ángel Jiménez Romero 1, Jorge Herrero Ciudad2 and Susana Martín
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Arroyo1
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1 Dpto. Ingeniería Eléctrica, Universidad de Zaragoza; [email protected]
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2 For Optimal Renewable Energy Systems, S.L. (4fores); [email protected]
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* Correspondence: [email protected]; Tel.: +34-976-761923
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Abstract: Cogging torque is a pulsating, parasitic and undesired torque ripple intrinsic of the
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design of a permanent magnet synchronous generator (PMSG), which should be minimized due to
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its adverse effects: vibration and noise. In addition, as aerodynamic power is low during start-up at
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low wind speeds in small wind energy systems, the cogging torque must be as low as possible to
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achieve a low cut-in speed. A novel mitigation technique using compound pre-slotting, based on a
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combination of magnetic and non-magnetic materials, is investigated. The finite element technique
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is used to calculate the cogging torque of a real PMSG design for a small wind turbine, with and
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without using compound pre-slotting. The results show that cogging torque can be reduced by a
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factor of 48% with this technique, while avoiding the main drawback of the conventional
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pre-slotting technique: the reduction of induced voltage due to leakage flux between stator teeth.
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Furthermore, through a combination of pre-slotting and other cogging torque optimization
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techniques, 84%, cogging torque can be eliminated for a given design.
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Keywords: cogging torque; permanent magnet synchronous generator; small wind turbines; finite
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element method; renewable energy, energy conversion.
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1. Introduction
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Increasing interest in the efficiency of electric machinery and reducing maintenance costs is
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making the use of permanent magnet synchronous generators (PMSGs) more common. PMSGs
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combine high efficiency with low maintenance and a high power density [1], factors that make them
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extremely attractive for use in renewable energy applications, such as wind [2], wave power [3] and
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tidal power [4], or electrical mobility applications [5] and, in general, in uses where they must act as
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a motor or generator. Furthermore, in renewable energy applications, PMSGs allow direct-drive
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configurations, making the use of gearbox unnecessary or reducing the number of gearbox stages,
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which decreases the overall generator volume and improves its efficiency [6].
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However, machines based on permanent magnets (PMs) also have some drawbacks, and the
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cogging torque is one of the main ones. The magnetic interaction between the flux generated by the
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rotor PMs and the stator geometry results in a pulsating torque called cogging torque, which,
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depending on the PM machine design, can cause an undesired ripple in both the machine’s induced
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voltage (EMF) and its mechanical torque [7,8]. Other problems with PMSGs are the vibrations and
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noise they make. Since this type of machine has high magnetic flux density values in the air gap, the
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electromagnetic forces between the PMs and the stator teeth are high [9]. These electromagnetic
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forces are divided into two components, one radial and the other tangential. The tangential
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component of the electromagnetic force contributes to the torque in the stator teeth, while the radial
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component causes vibrations and even deformations in the machine [10]. These radial forces act on
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the stator producing vibrations and noise, especially when their frequency coincides with the
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natural frequency of the machine’s mechanical structure [11].
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The cogging torque is especially important in wind energy applications as it establishes in which
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conditions the system will begin generating. The mechanical torque captured by the generation
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system must be larger than the cogging torque starting the rotation, which is why achieving a
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reduced cogging torque is one of the objectives for this type of machine.
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There are several methods to reduce cogging torque in the PMSG design phase. The most used
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is skewing, which consists of preventing the stator teeth and the magnets from becoming aligned by
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either turning the stator teeth [6, 12] or the rotor’s permanent magnets [1, 2, 13]. The required skew
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angle to largely cancel out the effect of the interactions between the PMs and the slots depends on
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how many slots and poles the machine has. Other methods study the use of notches in stator teeth
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[7, 14]. These notches produce the same effect in the magnetic interaction as the slots and increase
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the effective number of slots, which impacts on the cogging torque as it depends on how many poles
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and slots the machine has. Therefore, this method’s effectiveness is conditioned by the number of
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poles and slots selected in the design.
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A study is presented in [15, 16] in which slot openings in one half of the stator shift in one
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direction with respect to the tooth and the other half shift in the other direction. This means the
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cogging torque waveform moves in opposite directions in each machine half and the cogging caused
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by each machine half may be cancelled out depending on the shifted angle. Other studies focus on
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the shape of PM edges, concluding that their size can be reduced on the magnet sides to lessen air
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gap reluctance variation, which reduces the magnetic energy variation in the machine and,
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therefore, mitigates the cogging torque [17, 18].
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Several authors have conducted studies of PMSG with closed slots and their effects. Leakage
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fluxes caused as a result of closing stator slots are analyzed in [4], concluding that the size of PMs
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should be increased to compensate flux loss through closed slots. The increase in iron losses caused
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by tooth-tip saturation, distortion in the induced voltage this saturation causes and how the use of
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closed slots influences this are studied in [19 and 20]. The study by [21] focuses on average torque
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and its ripple in machines with closed slots for several stator types.
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Unlike the above-mentioned methods, which focus on minimizing the cogging torque in the
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machine design stage, this article proposes a cogging torque reduction method that is easy to
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implement without the need for any changes to the original design of the machine, a 6.3-kW
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generator for a small wind turbine. The suggested solution comprises sliding a metal part (pre-slot)
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into the slots after completing the machine winding. This technique minimizes slot openings so that
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induced voltage remains unaltered and the mounting of machine windings is not hampered. The
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results of the proposed method are analyzed using FEMM 2D finite element software on an original
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PMSG design and compared with the results obtained experimentally. Additionally, constructive
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improvements are suggested to reduce cogging torque. Finally, the article shows how the proposed
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technique can also be combined with the skewing technique, thus significantly reducing the cogging
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torque to 0.03 Nm in the ideal case and 0.51 Nm when imperfections in the manufacturing process
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2. Machine type and main parameters
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The machine involved in this study is a 6.3 kW PMSG with an interior rotor and
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surface-mounted magnets comprising 36 slots and 20 poles. Figure 1(a) is a cross section of the
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generator showing the slots forming the stator, the rotor in the internal part and the
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surface-mounted magnets above it in the central part. The details of the millimeter measurements of
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the machine’s slots and PMs are shown in Figure 2. The characteristic parameters of the studied
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PMSG are shown in Table 1.
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The analysis of the PMSG and the cogging torque reduction methods proposed in this study has
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been performed using FEMM 2D finite element software. To validate the FEMM model used in the
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cogging torque reduction analysis a comparison will be made with the experimental values of the
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original machine.
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(a) (b)
Figure 1. Cross section area of the PMSG: (a) Original model; (b) Model with centered holes.
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Figure 2. Slot and magnet dimensions (mm).
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Parameter Value
Phase 3
Pole number 20 Slot number 36 Rated speed 232 rpm Rated power 6300 W Rated voltage 256.4 V
Air gap 1 mm
Thickness of PM 3 mm Air gap diameter 188 mm
Material of steel M330-50A Material of PM NdFeB
3. Cogging torque
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Cogging torque is a parasitic torque resulting from interactions between the rotor’s permanent
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magnets and the stator slots. Air gap reluctance differs depending on the rotor’s angular position to
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the slots. Rotor magnets tend to align with the stator in the position in whichair gap permeance is
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larger [22], so when they are shifted from this position during rotation, they generate a torque, the
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cogging torque.
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Electromagnetic torque can be obtained from the variation in the total energy of the magnetic
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field compared with the angular position of the rotor θ when excitation current is constant [14]
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= − (1)
The total energy stored in the magnetic field or coenergy Wc in a PMSG is given by [7]
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=1
2 +
1
2( + ) ∅ + ∅ (2)
where L is the inductance of the windings, i the excitation current, R and Rm are, respectively, the
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reluctances viewed by the magnetomotive force and by the magnetic field, Фm the flux due to the
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magnets crossing the air gap, and N the number of winding turns.
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Therefore, substituting in (2) results in
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=1
2 −
1
2 ∅ +
∅ (3)
The second term of (3) corresponds to magnet reluctance torque and it is known as cogging
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torque [17], Tcog.
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= −1
2∅ (4)
As observed in (4), cogging torque is independent of the current and corresponds to the result
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of analyzing (3) when the machine is in open circuit. Cogging torque depends on magnetic flux and
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on the rate of change of air-gap reluctance. From (4), to minimize Tcog, reluctance R should be
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independent of the rotor position. Therefore, a very low cogging torque design requires an almost
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constant value of R for any rotor position.
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4. Cogging torque measurement
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Cogging torque is calculated in FEMM for every angular rotor position, making the machine
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operate off-load. The torque is calculated by integrating the Maxwell stress tensor throughout the air
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gap
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=
μ
(5)
where L is the rotor depth, g is the air gap length, Bn the normal flux density, Bt the tangential flux
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density and r the radius from the center of the rotor to the center of the air gap [7].
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Compared with the results obtained when the calculation is based on the magnetic energy
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variation with respect to the angular rotor position given by (1), in [18] it is shown that both methods
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obtain almost identical results.
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The simulation in FEMM of the PMSG in Figure 1 (a) obtained the cogging torque shown in
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Figure 3 (“original” curve), whose maximum value is 2.32 Nm, while the experimental results of the
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machine show maximum values of 3.70 Nm. The main reason for this deviation from experimental
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values is due to component manufacturing tolerance. Consequently, if a tolerance of ± 0.1 mm is
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included in the 20 PMs of the PMSG model and this error is distributed randomly at the height of the
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PMs, the result shown in Figure 3 (curve “with manufacturing errors”) is obtained. Having magnets
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that are not the same impacts the cogging torque significantly, mainly because differently sized PMs
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cause higher magnetic flux variations in the air gap. The maximum cogging torque value obtained in
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the simulation is 3.90 Nm (Figure 3). This value is slightly higher than the experimental PMSG
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results, making it possible to validate the developed FEMM model with respect to the cogging
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torque analysis.
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Figure 3. Simulation results of cogging torque of the original model considering manufacturing errors.
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5. Cogging torque reduction methods
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5.1.Pre-slot method
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The main objective is to reduce the cogging torque without affecting the machine’s construction
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characteristics and, therefore, without making any changes in the generator’s geometry.
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A closed-slot stator topology reduces reluctance variation in the air gap and, therefore, the
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machine’s maximum cogging torque value. Furthermore, the minimum dimensions of PMSG slot
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openings are conditioned by winding mounting factors. Their minimum size depends on the cross
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section of the winding conductors so that they can be inserted in the slot.
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Furthermore, the slot closing method has the drawback of generating a leakage flux through the
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slots due to the high permeability of the magnetic core connecting the teeth, Figure 4 (a). These
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leakages reduce the flux linked by the machine windings, thus producing a drop in induced voltage.
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This drop in induced voltage can be seen in Figure 5, showing the induced voltage of the PMSG with
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open slots (the “original” curve) and with closed slots. The effective induced voltage value is 256.4 V
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in the original generator model with open slots, and it decreases to 221.6 V when the slots are closed.
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(a) (b)
Figure 4. Magnetic field lines (θ=0°): (a) Model with closed slots; (b) Model with pre-slots.
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Figure 5. Effect of closed slots in back electromotive force (EMF).
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P
h
a
s
e
b
a
c
k
-E
M
F
(
V
To avoid the above-mentioned drawbacks, the proposed cogging torque reduction method
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consists of closing the slots by sliding in a pre-slot part made of the same ferromagnetic material as
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the stator, as observed in Figure 6. The pre-slot is placed between the teeth longitudinally after
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machine winding; this does not alter the winding or the slot fill factor. Figure 6 (a) provides details of
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the space between two of the machine’s stator teeth showing where the ferromagnetic part is slid
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into the start of each slot. The pre-slot considered in this study is 1.5 mm high; its dimensions are
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adjusted to the available space to render changing the machine winding unnecessary.
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A material separator with low magnetic permeability (aluminum or similar) and a width of
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1 mm, the same distance as the machine’s air gap, is in the central part of the pre-slot. The purpose of
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the central separator is to prevent the above-mentioned flux leakage linked by the windings. As this
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is a non-magnetic separator, it prevents the pre-slot from closing the magnetic field lines and,
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therefore, preventing flux from circulating between two consecutive PMs.
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(a) (b)
Figure 6. Proposed cogging-torque reduction method: (a) Pre-slot with separator; (b) Pre-slot with triangular
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separator.
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In accordance with the developed FEMM model, inserting pre-slots with a separator manages
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to reduce the maximum cogging torque value by 37.9% compared with the original PMSG, as
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observed in the results shown in the graph in Figure 7, but it does not decrease induced voltage as
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using the separator reduces leakages.
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This cogging torque reduction can be improved by considering other alternative geometrical
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pre-slot configurations. A triangular separator, as shown in Figure 6 (b), can lessen the magnetic
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energy variation caused in the original teeth edges or in pre-slot separators, which decreases the
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machine’s cogging torque.
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Pre-slot geometry with a triangular separator prevents leakage flux through it, as occurs with
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the central separator model, thus preventing the undesired decrease in induced voltage and in
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linked flux through machine windings. Similarly, it produces a higher reduction in maximum
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cogging torque, as shown in the graph in Figure 8, decreasing the cogging torque generated by over
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Figure 7. Comparison of cogging torque reduction with pre-slot method.
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Figure 8. Comparison of cogging torque reduction with the triangular pre-slot method.
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An analysis of induced voltage harmonics was conducted to confirm that the installation of the
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proposed pre-slot system to reduce cogging torque does not affect the machine’s technical
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characteristics. The PMSG is designed for small wind-power applications and, therefore, if an
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uncontrolled rectifier is used, the harmonics level is of no importance. However, if the connection is
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via a full converter, the opposite is the case. Figure 9 shows the frequency spectrum of the first 20
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harmonics of each of the waves and the harmonic distortion rate (THD) for each model; therefore,
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the analysis was conducted from the fundamental frequency of 38.67 Hz to the twentieth harmonic
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(773.4 Hz).
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Table 2 shows how the THD values obtained remain low and similar across all cases and never
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exceed 2%. The pre-slot solution with a triangular separator presents the least harmonic distortion of
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the considered models; it is very similar to the closed-slot model and improves the original
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configuration.
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Figure 9. Harmonic spectrum for the proposed reduction method.
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Table 2. THD of the different models.
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Model THD (%)
Original design 1.54 Design with closed slot 1.39 Design with pre-slot 1.88 Design with triangular pre-slot 1.33
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5.2.Manufacturing aspects to reduce the cogging torque
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Any change in the magnetic circuit alters its reluctance and, therefore, in accordance with (4), it
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affects the cogging torque and must be considered to reduce it. Consequently, it was found that the
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holes for correctly aligning the rotor sheet metal with screws in the original design significantly
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influence the machine’s cogging torque, depending on their position with respect to the PMs, Figure
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1(b).
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The impact of these holes on the cogging torque was analyzed for their different positions with
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respect to the PMs. The conclusion is that the optimal position, which minimizes cogging torque, is a
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centered position with respect to the magnets. Figure 10 (b) shows the case in which the rotor hole is
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centered with respect to the PMs. In this situation, the holes have virtually no influence on magnetic
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field lines linking one magnet with another. In contrast, when the hole is decentered, Figure 10 (a),
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the effect is a smaller effective area in the rotor through which the field lines circulate. Therefore,
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reluctance increases with respect to the case shown in Figure 10 (b) (centered hole) and the magnetic
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energy in the rotor decreases as observed in Figure 11.
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A
m
p
lit
u
d
e
(
%
(a) (b)
Figure 10. Magnetic field distribution (θ=0°): (a) Original model, without centered holes; (b) Model with
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centered holes.
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Figure 11. Magnetic energy in the rotor.
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Figure 12 shows the magnetic energy stored in the machine with respect to the rotor during
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electrical 360° (mechanical 36°) for both PMSG models. If the hole is decentered, energy minimums
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occur when the hole is aligned with a stator tooth. In this position, the flux between two adjacent
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magnets would be the maximum if there was no hole. Consequently, as observed in Figure 12, and
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given that the teeth are distributed every mechanical 10° along the stator, the energy minimum
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occurs with this frequency.
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E
n
e
rg
y
(
J
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Figure 12. Total magnetic energy.
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Figure 13 compares the cogging torque for both PMSGs, resulting in a higher value when the
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hole is decentered given that more magnetic energy variations occur, as presented in Figure 12. The
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cogging torque of the model with decentered holes is 2.32 Nm, while this value does not exceed
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0.86 Nm when the holes are centered. Figure 13 also shows that the presence of decentered holes
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even produces a change in the cogging torque wave period.
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Figure 13. Cogging torque of the original PMSG and model with centered holes.
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245
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40 45 50 55 60 65 70
Angular position (º) 45.9
45.92 45.94 45.96 45.98 46 46.02 46.04 46.06
46.08 Total energy
5.3.Comparative results
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Table 3 compares the maximum cogging torque values obtained with the original design with
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the two proposed pre-slot configurations and shows the reduction percentage with respect to the
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original design. The results of the different models are shown in the ideal case (with no
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manufacturing errors) and considering manufacturing errors in the magnets.
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Table 3. Comparison of the maximum cogging torque values obtained for the prototype and the different
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models considered in the study.
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Model Without manufacturing errors Considering manufacturing errors
Nm Reduction (%) Nm Reduction (%)
Original design (without centered holes)
Prototype 3.70
Original design 2.32 - 3.92 -
Pre-slot with separation 1.44 37.9 2.03 48.2
Triangular pre-slot 1.21 47.8 1.90 51.5
Design with all holes centered
Original design 0.86 - 3.31 -
Pre-slot with separation 0.61 29.1 1.80 45.6
Triangular pre-slot 0.59 31.4 1.76 46.8
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Finally, the proposed pre-slot method was compared with the skewing technique and the
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combination of both is considered. The technique of fractional skewing in the rotor [23] comprises
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dividing the rotor and turning one division away from another for half the cogging torque period.
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Four divisions have been considered in this analysis, as observed in Figure 14. Therefore,
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considering that the cogging torque period is mechanical 2, the shift of one division with respect to
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another is half of this period, which equals 1. As observed in Table 4, concerning the model with
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centered holes and in the ideal case of having no manufacturing errors, applying this combined
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technique manages to reduce the cogging torque to a peak value of 0.03 Nm or to 0.51 Nm if
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manufacturing errors are considered. In either of the two cases, the cogging torque reduction is very
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significant.
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Figure 15 shows the cogging torque waveform obtained for the PMSG with centered holes.
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Because of the reluctance periodicity, the cogging torque is a periodic waveform with a frequency
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given by:
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= ∙ ,
360° = 696
(6)
where is the mechanical speed (1392°/s), LCM the least common multiple of the number of slots
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(Nslots=36) and the number of poles (Npoles=20). The results in Figure 15 show the decrease in the
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cogging torque with the pre-slot triangular method and that this improvement is even better when
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combining this pre-slot installation technique with fractional skewing in the rotor, up to 84% less in
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Figure 14. Fractional skewing in the rotor.
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Table 4. Comparison of the maximum cogging torque values of PMSG with centered holes and the different
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models considering skewing.
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Model Without manufacturing
errors
Considering manufacturing errors Nm Reduction (%) Nm Reduction (%)
Design with centered holes 0.86 - 3.31 -
Design with centered holes + Triangular
pre-slot 0.59 31.4 1.76 46.8
Design with centered holes + Skewing 0.31 64.0 1.34 59.5 Design with centered holes + Triangular
pre-slot + Skewing 0.03 96.5 0.51 84.6
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Figure 15. Cogging torque of the centered holes PMSG model, triangular pre-slot model and triangular pre-slot
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model with skewing (with manufacturing errors).
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6. Conclusions
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This article presents a new cogging torque reduction technique that does not require changes to
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the machine’s main geometry. It proposes placing pre-slots in the initial part of the stator slots. These
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pre-slots are made of the same ferromagnetic material as the stator, with a non-magnetic central
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separator (in two halves). The pre-slots are slid longitudinally between the slots after completing
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machine winding and, therefore, without altering the PMSG’s fill factor.
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Introducing a central part of non-magnetic material prevents leakage flux between the
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machine’s teeth and also stops its induced voltage from reducing significantly with respect to the
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configuration without pre-slots.
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The proposed method manages to reduce cogging torque in PMSGs with surface-mounted
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magnets by up to 47.8%. Additionally, the article analyzes how changing the magnetic circuits for
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construction reasons can affect the cogging torque, which can easily be optimized. The pre-slot
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technique is also compatible with other cogging torque reduction techniques, such as skewing.
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When the above-mentioned methods are combined, cogging torque is reduced by 84.6% considering
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manufacturing errors.
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Author Contributions: Miguel García-Gracia and Ángel Jiménez Romero performed the simulations and wrote
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the paper. 4fores contributed with the prototype and valuable comments and corrections. All authors discussed
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the results and commented on the manuscript at all stages.
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Funding: This research was funded by “Ministerio de Economía, Industria y Competitividad (Plan Estatal de
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Investigación Científica y Técnica y de Innovación 2013-2016)” and “Fondo Europeo de Desarrollo Regional FEDER”,
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grant number RTC-2016-5234-3, in the frame of the Project MHiRED “Nuevas tecnologías para minirredes híbridas
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eólica-fotovoltaica gestionadas con almacenamiento en conexión a red y con apoyo síncrono en funcionamiento en isla”.
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Acknowledgments: The authors wish to thank 4fores for granting their permission to publish some data
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presented in this article. Furthermore, the technical support from the Research Group on Renewable Energy
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Integration of the University of Zaragoza (funded by the Gobierno de Aragón) is also gratefully acknowledged.
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Conflicts of Interest: The authors declare no conflict of interest.
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